A305807 Dirichlet inverse of A032742 (the largest proper divisor of n).

1, -1, -1, -1, -1, -1, -1, -1, -2, -3, -1, 1, -1, -5, -3, -1, -1, 0, -1, 1, -5, -9, -1, 3, -4, -11, -4, 1, -1, 5, -1, -1, -9, -15, -5, 6, -1, -17, -11, 5, -1, 7, -1, 1, -2, -21, -1, 5, -6, -8, -15, 1, -1, 4, -9, 7, -17, -27, -1, 19, -1, -29, -4, -1, -11, 11, -1, 1, -21, -3, -1, 8, -1, -35, -8, 1, -9, 13, -1, 9, -8, -39, -1, 29, -15, -41, -27, 11, -1

Offset: 1

Author

Antti Karttunen, Jun 13 2018

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A032742, A300236, A300239, A305808.

Programs

Mathematica

b[n_] := If[n == 1, 1, Divisors[n][[-2]]];
a[n_] := a[n] = If[n == 1, 1, -Sum[b[n/d] a[d], {d, Most@ Divisors[n]}]];
Array[a, 100] (* Jean-François Alcover, Feb 17 2020 *)

PARI

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A305807(n) = if(1==n,1,-sumdiv(n,d,if(dA032742(n/d)*A305807(d),0)));

Formula

a(1) = 1; for n > 1, a(n) = -Sum_{d|n, dA032742(n/d)*a(d).